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Tensor Networks and You Nikko Pomata How do you network tensors? Tensors: a review The tensor-network notation T ENSOR N ETWORKS Tensor network examples and You Tensor network methods Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization Nikko Pomata Tensor Networks and You Stony Brook University Grad Talks: April 25 2018

Tensor Networks O UTLINE and You Nikko Pomata How do you H OW DO YOU NETWORK TENSORS ? network tensors? Tensors: a review Tensors: a review The tensor-network notation Tensor network examples The tensor-network notation Tensor network methods Tensor network examples Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors T ENSOR NETWORK METHODS Entanglement Renormalization Matrix product states (MPS) Tensor Networks and You Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization T ENSOR N ETWORKS AND Y OU : F RONTIERS IN T ENSOR N ETWORK R ESEARCH

Tensor Networks R EMINDER : W HAT ARE TENSORS , ANYWAY ? and You Nikko Pomata Basic idea How do you network tensors? A tensor is a linear combination of tensor products of Tensors: a review vectors: The tensor-network notation Tensor network examples Tensor network v ( j ) 1 ⊗ v ( j ) ⊗ · · · ⊗ v ( j ) � T = methods n 2 Matrix product states (MPS) j Projected Entangled Pair States � Coarse-graining tensors = T i 1 , i 2 ,..., i n e i 1 ⊗ e i 2 ⊗ · · · ⊗ e i n Entanglement Renormalization i 1 , i 2 ,..., i n Tensor Networks and You in terms of basis vectors More formally: A tensor is a multilinear map on vector spaces � · · · T i 1 ,..., i m j 1 ,..., j n = � T e i 1 � e i m � � � e j 1 � � � e j n � � � · · ·

Tensor Networks R EMINDER : W HAT ARE TENSORS , ANYWAY ? and You Nikko Pomata How do you network tensors? Tensors: a review The tensor-network notation Tensor network examples Tensor network methods A tensor is not Matrix product states (MPS) Projected Entangled Pair � A quantity that transforms covariantly with coordinate States Coarse-graining tensors changes (a tensor fi eld ) Entanglement Renormalization � An observable that has multiple indices which Tensor Networks and You transform under SO ( 3 ) rotations (a tensor operator )

Tensor Networks T ENSOR P RODUCTS IN Q UANTUM P HYSICS and You Nikko Pomata How do you network tensors? Tensors: a review If a quantum system is comprised of two subsystems A The tensor-network notation Tensor network examples and B , represented by Hilbert spaces H A and H B , then Tensor network the overall Hilbert space is H A ⊗ H B methods Matrix product states (MPS) Projected Entangled Pair States These subsystems can be: Coarse-graining tensors Entanglement Renormalization � The states of two different particles Tensor Networks and You � The position of a particle along the x axis, and the position of the same particle along the y axis � A particle’s position, and the same particle’s spin � The number of bosons in each of two different states

Tensor Networks A M ANY - BODY S PIN S YSTEM and You Nikko Pomata How do you network tensors? Tensors: a review The tensor-network notation Tensor network examples Often wind up studying lattice systems where every Tensor network methods lattice site is a fi nite-level system (e.g. the spin of an Matrix product states (MPS) Projected Entangled Pair atom at that site) States Coarse-graining tensors Entanglement Renormalization If there are N sites represented by C d : then Tensor Networks and You C d � ⊗ N � H =

Tensor Networks D RAWING T ENSORS and You Nikko Pomata In tensor-network notation, a tensor is drawn as a shape and its indices are drawn as lines: How do you network tensors? Tensors: a review The tensor-network notation i Tensor network examples Tensor network i methods Matrix product states (MPS) Projected Entangled Pair States A ijkl = j l Coarse-graining tensors v ijk = Entanglement Renormalization k Tensor Networks j k and You j i u iji � j � = i � j �

Tensor Networks P UTTING T OGETHER T ENSOR N ETWORKS and You Nikko Pomata How do you A tensor network is linked together by contractions (just network tensors? Tensors: a review like in GR) The tensor-network notation Tensor network examples Tensor network k methods Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors l Entanglement r i Renormalization � u ipjq A krpl v rqm p = Tensor Networks and You pqr j q m

Tensor Networks E XAMPLE : T HE I SING P ARTITION F UNCTION and You Nikko Pomata   How do you network tensors? � � � �  = e − β s i s j Z ( β ) =  − β exp s i s j Tensors: a review The tensor-network notation { s i } � i , j � { s i } � i , j � Tensor network examples Tensor network methods Matrix product states (MPS) Projected Entangled Pair States s Coarse-graining tensors Entanglement = e − β ss � Renormalization Tensor Networks and You s � s a = δ s a , s b , s c , s d s d s b s c

Tensor Networks E XAMPLE : T HE I SING P ARTITION F UNCTION and You Nikko Pomata   How do you network tensors? � � � �  = e − β s i s j Z ( β ) =  − β exp s i s j Tensors: a review The tensor-network notation { s i } � i , j � { s i } � i , j � Tensor network examples Tensor network To obtain � s m s n � : methods Matrix product states (MPS) Projected Entangled Pair States s Coarse-graining tensors Entanglement = e − β ss � Renormalization Tensor Networks and You s � s a = δ s a , s b , s c , s d s d s b s c s a = s a δ s a , s b , s c , s d s d s b s c

Tensor Networks M ANIPULATING T ENSORS : U NITARIES AND and You I SOMETRIES Nikko Pomata Often wind up dealing with unitary and isometric tensors: How do you network tensors? Tensors: a review The tensor-network notation j i Tensor network examples u u † Tensor network = δ ii � δ jj � = = methods Matrix product states (MPS) Projected Entangled Pair u u † States Coarse-graining tensors i � j � Entanglement Renormalization Tensor Networks and You w † w = , = P w † w

Tensor Networks M ANIPULATING T ENSORS : SVD and You Nikko Pomata The singular value decomposition: How do you network tensors? Tensors: a review A = USV † The tensor-network notation Tensor network examples Tensor network � A is an arbitrary matrix methods Matrix product states (MPS) � S is diagonal Projected Entangled Pair States Coarse-graining tensors � U and V are unitary Entanglement Renormalization In order to turn a matrix equation into a tensor equation, Tensor Networks and You group indices: = A V † U S

Tensor Networks and You Nikko Pomata H OW DO YOU NETWORK TENSORS ? How do you Tensors: a review network tensors? Tensors: a review The tensor-network notation The tensor-network notation Tensor network examples Tensor network examples Tensor network methods Matrix product states (MPS) Projected Entangled Pair T ENSOR NETWORK METHODS States Coarse-graining tensors Matrix product states (MPS) Entanglement Renormalization Projected Entangled Pair States Tensor Networks and You Coarse-graining tensors Entanglement Renormalization T ENSOR N ETWORKS AND Y OU : F RONTIERS IN T ENSOR N ETWORK R ESEARCH

Tensor Networks O BTAINING THE M ATRIX P RODUCT S TATE and You Nikko Pomata How do you network tensors? Tensors: a review � Start with any state on a spin chain ( C d ⊗ N ) The tensor-network notation Tensor network examples � Apply SVD to the state to divide it into two parts Tensor network methods Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization Tensor Networks and You

Tensor Networks O BTAINING THE M ATRIX P RODUCT S TATE and You Nikko Pomata How do you network tensors? Tensors: a review � Start with any state on a spin chain ( C d ⊗ N ) The tensor-network notation Tensor network examples � Apply SVD to the state to divide it into two parts Tensor network methods Matrix product states (MPS) Projected Entangled Pair States Coarse-graining tensors Entanglement Renormalization Tensor Networks and You

Tensor Networks O BTAINING THE M ATRIX P RODUCT S TATE and You Nikko Pomata How do you network tensors? Tensors: a review � Start with any state on a spin chain ( C d ⊗ N ) The tensor-network notation Tensor network examples � Apply SVD to the state to divide it into two parts Tensor network methods � ... then keep applying SVD to remaining parts to Matrix product states (MPS) Projected Entangled Pair divide the state site by site States Coarse-graining tensors Entanglement Renormalization Tensor Networks and You